Equivalence of two definitions of Whitehead Torsion

Question

In their book Lecture Notes in Algebraic Topology, Davis and Kirk define the torsion of an acyclic chain complex $C$ in the following way:

Since $C$ is acyclic, there exists a simple chain complexes $E,F$ and a chain isomorphism $f: E \rightarrow F \bigoplus C$ (a simple chain complex is a finite direct sum of elementary chain complexes which have only two non-zero terms and the identity map between these two terms). Then define the torsion $\tau(C)$ of $C$ as $\sum_i (-1)^n [f_n: E_n \rightarrow F_n \bigoplus C_n] \in \tilde{K_1}(R)$, which can be shown to be independent of $E,F$. There is a copy of the book at http://www.indiana.edu/~lniat/book.pdf from Kirk's website which contains this material in Chapter 11.

Another definition of the torsion is as follows: Let $C$ be an acyclic chain complex with $C_n$ a free $R$ module with specified basis. Then there exists a chain homotopy $s$ (which is a sequence of maps $s_n: C_n \rightarrow C_{n+1}$) between the identity and zero map. Then $d + s$ is an isomorphism between the $R$-modules $C_{odd}$ and $C_{even}$. Then we can consider $[d+s] \in \tilde{K_1}(R)$. This is the same as the wikipedia definition of Whitehead torsion at http://en.wikipedia.org/wiki/Whitehead_torsion.

I was wondering if someone can help me see why these two definitions are the same. Also, Exercise 198 of the Davis/Kirk book is to show that these two definitions are the same. I was able to use $s$ to construct explicit $E,F$ and a map $f$ between $E$ and $F \bigoplus C$ but I did not get equivalence of the two definitions. I got the following modules for $E$ and $F$:

$E_n = C_1 \bigoplus C_2 \bigoplus C_3 \bigoplus ... \bigoplus C_n$

$F_n = C_1 \bigoplus C_2 \bigoplus C_3 \bigoplus ... \bigoplus C_{n-1}$.

Answer 1

We have that the Whitehead torsion is the unique function $\tau$ from the class of acyclic based complexes to $\tilde{K_1}(R)$ satisfying the following three properties.

1. If $f:C\to C'$ is a simple isomorphism (i.e., a chain isomorphism such that for all $i$, $[f_i]:=$ the image of $f_i:C_i\to C'_{i}$ in $\tilde{K_1}(R)$ corresponds to the identity element), then $\tau(C)=\tau(C')$.
2. $\tau(C\oplus C') = \tau(C)+\tau(C')$.
3. $\tau(0\to C_n \xrightarrow{f} C_{n+1}\to 0) = (-1)^{n-1}[f]$ for any isomorphism $f$ of based modules $C_n$ and $C_{n+1}.$

The uniqueness is proven as $(17.3)$ in Marshall M. Cohen's "a course in simple-homotopy theory".

Both of the two definitions satisfy this characterization: in Davis-Kirk this is Theorem $11.12$, and for $[d+s]$ this is $(17.1)$ of Cohen.

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A quicker way for a reader of Davis-Kirk is to use $(14.2)$ of Cohen, which is needed in the proof of Cohen's $(17.3)$ anyway. It states that after direct summing with elementary (or, simple, in the terminology of Cohen) chain complexes $T,T'$, there exists a simple isomorphism $C\oplus T \xrightarrow{\sim} C'\oplus T'$, where $$ C' = 0 \to C'_n \xrightarrow{f} C'_{n-1} \to 0 $$ for some odd $n$ with $C'_n = C_{\text{odd}}$, $C'_{n-1}=C_{\text{even}}$ and $f = (d+s)|_{C_\text{odd}}$. By $1,2,3$ in the above characterization proven as Theorem $11.12$ in Davis-Kirk, we see that according to Davis-Kirk's definition, $\tau(C) = [[d+s)_\text{odd}]$.